Defining polynomial
|
\(x^{12} + 4 x^{11} + 8 x^{7} + 4 x^{2} + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $34$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[3, \frac{23}{6}]$ |
| Visible Swan slopes: | $[2,\frac{17}{6}]$ |
| Means: | $\langle1, \frac{23}{12}\rangle$ |
| Rams: | $(6, 11)$ |
| Jump set: | $[3, 9, 21]$ |
| Roots of unity: | $2$ |
Intermediate fields
| 2.1.3.2a1.1, 2.1.6.11a1.11 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{12} + 4 x^{11} + 8 x^{7} + 4 x^{2} + 2 \)
|
Ramification polygon
| Residual polynomials: | $z^8 + z^4 + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $2$,$1$,$1$ |
| Indices of inseparability: | $[23, 12, 0]$ |
Invariants of the Galois closure
| Galois degree: | $1536$ |
| Galois group: | $C_2\wr S_4$ (as 12T223) |
| Inertia group: | $C_2\wr A_4$ (as 12T187) |
| Wild inertia group: | $C_2^2:D_4^2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3, \frac{23}{6}, \frac{23}{6}]$ |
| Galois Swan slopes: | $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2,\frac{17}{6},\frac{17}{6}]$ |
| Galois mean slope: | $3.5807291666666665$ |
| Galois splitting model: | $x^{12} + 2 x^{10} + x^{8} - 9 x^{4} - 22 x^{2} - 11$ |